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INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL

J. Phys. A: Math. Gen. 35 (2002) 4453–4476 PII: S0305-4470(02)29341-5

An accurate method for numerical calculations inquantum mechanics

Hideaki Ishikawa

Fujitsu Laboratories Ltd, 50 Fuchigami, Akiruno, Tokyo 197-0833, Japan

Received 1 October 2001, in final form 23 January 2002Published 10 May 2002Online at stacks.iop.org/JPhysA/35/4453

AbstractAn accurate method for numerical calculations of matrix elements and forsolving the eigenvalue problem in quantum mechanics is presented. Methodsfor numerical interpolation, differentiation and integration provide 15-digitaccuracy with double-precision arithmetic operations. A method for solutionof the eigenvalue problem of an ordinary differential equation by usingdiscretization and matrix eigenvalue methods provides 13- to 15-digit accuracy.The efficiency of the proposed methods is demonstrated by the applicationsto bound states for the linear harmonic oscillator, anharmonic oscillators, theMorse potential and the modified Poschl–Teller potential.

PACS numbers: 02.60.−x, 02.70.−c, 03.65.−w, 31.15.−p

1. Introduction

Since the development of wave mechanics [1] the Schrodinger equation has been appliedto many fields in quantum mechanics [2–6]. Exact analytic solutions with special or othermathematical functions are obtained for limited cases of, for example, a free particle, linearharmonic oscillator and hydrogen atom. Approximation methods for solutions, such asperturbation, variation and Wentzel–Kramers–Brillouin (WKB), have been extensively usedbut their applicable range is rather restricted for practical problems. In order to overcomethese limitations, numerical methods of solution by matching or shooting wavefunctionsobtained by the Numerov method have been developed for atomic structure calculations sincethe early days of wave mechanics [7–11]. Though pioneering works have provided fruitfulinsight into atomic structure calculations, errors in numerical calculations are usually largerthan 1.0D-8 (= 1.0 × 10−8) for eigenvalues, so that further improvements in accuracy arenecessary. The approach via shooting for eigenvalues and expectation values, and usingRichardson extrapolation for eigenvalues, has been proposed in [12–20] but provides nodescription of the accuracy of the eigenfunctions, off-diagonal matrix elements between

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4454 H Ishikawa

different eigenfunctions or matrix elements such as derivatives. Another method of solution isthe discretized matrix eigenvalue problem [21–25]. Though the higher-order finite differenceformulae for the second derivative have been used, the eigenvalues had only 8-digit accuracy.Accurate calculation of eigenvalues and eigenfunctions, and of matrix elements, is basic to awide range of applications of quantum mechanics.

The main theme of the present paper is highly accurate calculations for quantum mechanicsusing the simplest possible methods. They are very useful in large-scale computations inatomic and molecular physics. Our final aim is to calculate matrix elements such as totalenergies, transition probabilities in photoabsorption and photoelectron spectra of atoms andmolecules; high accuracy is required for the case where small matrix elements play animportant role, for example, inelastic collisions between charged particles and molecules.In previous papers [26, 27] we presented a method of accurate single-centre and multicentrenumerical integration and demonstrated its applicability to atomic structure and molecularorbital calculations, respectively. Matrix elements for atomic structure calculations with15-digit accuracy, which is the highest accuracy achieved in double-precision arithmeticoperations, were obtained by using Gaussian quadrature rules for the exact analytic solutionof a hydrogenic basis function. Matrix elements with 10-digit accuracy were obtained forthe numerical basis set, in which the wavefunction is given in a tabular form at discretepoints, and functions at other points are calculated by interpolation. The topics needingimprovement for these investigations are the accuracy relating to the integrand, becausea loss of significant digits occurs frequently during a process involving a large numberof computations. The first topic for improvement is interpolation with 10-digit accuracybecause it is used ubiquitously. The second topic is numerical derivatives. The calculationof derivatives has been generally avoided because adequate accuracy has not been achieved.This, however, restricts the applicability of the numerical calculations. The third topic isnumerical integration. In addition to Gaussian quadrature rules, accurate numerical integrationusing functions only at tabular points is frequently used during the process of calculation.The fourth topic is improvement in the accuracy of eigenvalues and eigenfunctions of theSchrodinger equation. Although a large number of references on numerical analysis [28–37],interpolation [38–40], numerical derivatives [28, 29, 31, 41, 42], integration [42–47], the two-point boundary-value problem of ordinary differential equations [48–50] and the eigenvalueproblem of the Schrodinger equation [12–25] show formulae and typical examples ofcalculations, the ultimate performance of the calculation method has not yet been fullyinvestigated. It is a challenging and non-trivial problem to overcome the loss of accuracyin these arithmetic operations. After trials for improving accuracy we found that the classicalmethods, such as Lagrange interpolation, numerical derivatives, central-difference integrationformula and finite difference methods, provide accurate results. Though they are well known,their ultimate performance has not been well recognized. Since the classical methods aresimple, they serve as powerful tools across a wide range of research areas. Since theproblem is general, we summarize the relation between quantum mechanics and numericalcalculations in section 2. In section 3 we present the calculation method. Section 4 isdevoted to results and discussion. We take examples from the one-dimensional potentialproblem, the linear harmonic oscillator [1–6,51], anharmonic oscillators of the potential v(ξ) =µξ 2 + λξ 4 [12,14,15,18,19,23,24,52–70] and V (ξ) = ξ 2 + λξ 2/(1 + gξ 2) [18,23,24,71–79],the Morse potential [2–6,19,22,25,51,80–84] and the modified Poschl–Teller potential [3–5]in order to clearly demonstrate the performance of our method, as accurate solutions by othertechniques exist in many fields of physics and chemistry. Applications to atomic structurecalculations will be published elsewhere. In the appendix, the central-difference integrationformula of high degree is derived.

An accurate method for numerical calculations in quantum mechanics 4455

2. Relation between quantum mechanics and numerical calculation

The one-electron Hamiltonian of quantum mechanics in one dimension is given by

H = −(h2/2m)(d/dx)2 + U(x) (1)

where the first term is a kinetic energy with mass m and Dirac constant h and the second termis the potential energy. The wavefunction ψν belonging to the eigenenergy Eν of the quantumnumber ν satisfies the Schrodinger equation

Hψν = Eνψν (2)

with suitable boundary conditions. Thus the eigenvalue problem in quantum mechanics is theboundary-value problem of the second-order ordinary differential equation.

With the wavefunctions we can calculate matrix elements of the operator A =A(x, d/dx, (d/dx)2):

〈ν|A|ν ′〉 =∫ ∞

−∞dx ψν(x)Aψν ′(x). (3)

The matrix elements are, for example, the orthonormal integral 〈ν|ν ′〉 = δνν ′ for the operatorA = 1, position 〈ν|x|ν ′〉, momentum 〈ν| − ih(d/dx)|ν ′〉, potential energy 〈ν|U(x)|ν ′〉 andkinetic energy 〈ν| − (h2/2m)(d/dx)2|ν ′〉. The orthonormal integral with analytic exactwavefunctions provides a check on the accuracy of numerical integration and on the accuracyof the integrand calculated using the interpolation. The matrix elements of the position andpotential energy also give another check on numerical integration. The matrix elements of themomentum and kinetic energy give a check on the first and second derivatives calculated byusing numerical differentiation. The relation between the eigenvalue and expectation value ofthe Hamiltonian also gives another check on the accuracy of the matrix elements:

Eν = 〈ν|H |ν〉/〈ν|ν〉 = 〈ν| − (h2/2m)(d/dx)2 + U(x)|ν〉/〈ν|ν〉. (4)

In evaluating the matrix elements it is important to accurately calculate integrals and integrandsat the tabular points and at the intermediate points between the tabular points. In the following,we proceed to calculate accurate interpolation, numerical derivatives, numerical integrationand to solve the eigenvalue problem of ordinary differential equations.

3. Numerical methods of calculation

3.1. Interpolation

Let the mesh points for x be taken equidistant along the linear scale x and a function y = f (x)

be given in a tabular form at these discrete points. Let us take a function y = f (x) at (n + 1)

points yk = f (xk), k = 0, 1, 2, . . . , n, where xk is arranged in increasing order, and let uscalculate the function f (x) at an intermediate point by interpolating between these points.Since the interpolation is used ubiquitously in numerical calculations, it should be simple andaccurate. Though methods of interpolation have long been known, their accuracy has not beencarefully studied. We demonstrate here that a method of interpolation satisfying the aboverequirements is the classical Lagrangian interpolation [28, 29, 33, 38–40], where the functionis approximated by a polynomial pn(f, x) of degree n:

f (x) = pn(f, x) + Rn (5)

where

pn(f, x) =n∑

k=0

�k(x)f (xk) (6)

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and the polynomials �k(x) of degree n are the Lagrangian interpolation coefficients:

�k(x) =n∏

j=0,j �=k

(x − xj )/(xk − xj ). (7)

The remainder term Rn is given by

Rn = πn(x)f [x0, x1, . . . , xn, x] ≈ πn(x)f (n+1)(ξ)/(n + 1)! (8)

πn(x) =n∏

i=0

(x − xi), (9)

where f [x0, x1, . . . , xn, x] denotes the divided difference [28, 38–40] and f (n+1)(ξ) is the(n+1)th derivative at x0 � ξ � xn. In the case of equidistant intervals, that is, h = (xn−x0)/n,the truncation error depends on h, n and f (n+1)(ξ). If the function f (x) is continuous andsmooth, the truncation error is reduced by increasing n, with a suitable choice of h determinedby numerical experiments as described later, though the use of lower-order interpolationpolynomials has often been recommended [37]. The classical method with Lagrangianinterpolation polynomials is also useful if there are many interpolated functions at the samepoint because the Lagrangian interpolation coefficient at that point is calculated only once andcan be used repeatedly. This is often the case in physical problems. The accurate interpolationenables us to calculate numerical derivatives and integrals by using higher-order interpolationpolynomials.

3.2. Numerical derivative

The numerical derivative at any point can be calculated in two steps. First, the derivatives atthe tabulated points are calculated by using the derivative formula obtained by differentiatingthe Lagrangian interpolation formula (6) and evaluating the derivatives at the tabular points.For the equidistant interval h, the mth derivative of y = f (x) at xk, k = 0, 1, . . . , n, is givenin the form

[(d/dx)my]k = (m!/hm)

[(1/n!)

n∑j=0

mnAkjyj + mnEk

](10)

where the coefficients mnAki and the truncation errors mnEk up to n = 10 are tabulatedin [29, 41]. The formula at the central point is mainly used because the truncation error isminimum. We obtain the formula at the centre xi of the (n + 1) points for n = 12 given by

[(d/dx)2y]i = (1/h2)(1/831 600)[−50yi−6 + 864yi−5 − 7425yi−4 + 44 000yi−3

− 222 750yi−2 + 1425 600yi−1 − 2480 478yi

+ 1425 600yi+1 − 222 750yi+2 + 44 000yi+3

− 7425yi+4 + 864yi+5 − 50yi+6] + O(h14) (11)

and for n = 14

[(d/dx)2y]i = (1/h2)(1/75 675 600)[900yi−7 − 17 150yi−6 + 160 524yi−5

− 1003 275yi−4 + 4904 900yi−3 − 22 072 050yi−2 + 132 432 300yi−1

− 228 812 298yi + 132 432 300yi+1 − 22 072 050yi+2 + 4904 900yi+3

− 1003 275yi+4 + 160 524yi+5 − 17 150yi+6 + 900yi+7] + O(h16). (12)

The formulae at the non-central point are exceptionally used at points near the edge of the wholeinterval, where the central point formula cannot be used. We can calculate the second derivative


either by using the expression for m = 2 in equation (10) or by using the first derivative twice;both methods are useful. The derivatives can be calculated accurately by choosing the degreen and the width of the interval h as described later. Second, the derivatives at points otherthan tabulated ones are obtained by interpolating between derivatives at tabular points usingthe Lagrangian interpolation method as described earlier. Since the Lagrangian interpolationis accurate, errors in the derivatives come from those at the tabular points.

In evaluating the derivatives, the error of the numerical derivative consists of the truncationerror (m!/hm)mnEk and the round-off error [29, 34, 37]. The truncation error of the firstderivative at the tabular point is given in the form cn,1h

nf (n+1)(ξ) and that of the secondderivative at the central point is cn,2h

nf (n+2)(ξ), where cn,1 and cn,2 are coefficients thatare decreasing functions of n. The round-off errors for the first and second derivatives areproportional to 1/h and 1/h2, respectively.

3.3. Numerical integration

Accurate numerical integration by evaluating functions only at the tabular points can beachieved by using the central-difference integration formula [28]. Let us take a functiony = f (x) at discrete and distinct (n + 1) points yk = f (xk), centred at xi , k = i − (n/2),i − (n/2)+1, . . . , i −1, i, i +1, . . . , i + (n/2), where n is an even number and xk is arranged inincreasing order. The integration formula over the three central points with interval [xi−1, xi+1]is given in the form∫ xi+1

xi−1

f (x) dx = h[Bi−(n/2)fi−(n/2) + Bi−(n/2)+1fi−(n/2)+1 + · · · + Bi−1fi−1 + Bifi

+ Bi+1fi+1 + · · · + Bi+(n/2)−1fi+(n/2)−1 + Bi+(n/2)fi+(n/2)] + O(hn+2) (13)

where Bk are constants. Here, in addition to the functions within the interval, the functionsoutside the interval are also used for evaluating the integral [39]. The integration formulae forn = 2 (Simpson’s rule) and 4 have been shown in [9]. We obtain new integration formulae,with small truncation errors, for n = 6 and 8 given, respectively, by∫ xi+1

xi−1

f (x) dx = (h/3780)[5f i−3 − 72fi−2 + 1503fi−1 + 4688fi

+ 1503fi+1−72f i+2 + 5fi+3] + O(h9) (14)

and∫ xi+1

xi−1

f (x) dx = (h/113 400)[−23f i−4 + 334fi−3 − 2804fi−2 + 46 378fi−1

+ 139 030fi + 46 378fi+1 − 2804fi+2 + 334fi+3 − 23fi+4] + O(h11). (15)

The numerical integration over the whole interval can be performed by repeated use of theseformulae.

Another accurate method of numerical integration over the whole interval (−∞, ∞) witha small number of integration points is the Gauss–Hermite quadrature rule:

∫ ∞

−∞f (x) dx =

∫ ∞

−∞exp(−x2)F (x) dx =

n∑k=1

ωkF(xk) (16)

where

F(x) = f (x) exp(x2) (17)

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and where xk are zeros of the Hermite polynomials Hn(x) of degree n and ωk is a weight ofthe Gauss–Hermite quadrature rule [42–47]:

Hn(x) = (−1)nexp(x2)(d/dx)nexp(−x2) = 2xHn−1(x) − 2(n − 1)Hn−2(x) (18)

ωk = 2n+1n!π1/2/[Hn+1(xk)]2. (19)

The function at xk values that are usually different from the tabular points can be calculatedby using the accurate interpolation formula in section 3.1.

3.4. Numerical solution of the eigenvalue problem of ordinary differential equations by usingthe matrix eigenvalue method

The differential equation can be transformed into the matrix eigenvalue problem by applyingdiscretization in space coordinates. By using the formula for a second-order derivative at thecentre xi of the (n + 1)-discretized points:

[(d/dx)2y]i= anyi+(n/2) + an−1yi+(n/2)−1 + · · · + a(n/2)+1yi + · · · + a0yi−(n/2) (20)

where ak = (2/h2)(1/n!)2,nA(n/2),k , the differential equation is written as a matrix eigenvalueequation:

AY = EY (21)

where

A =

a(n/2) + U0 a(n/2)−1 · · · a0 0 0 · · · · · ·a(n/2)+1 a(n/2) + U1 a(n/2)−1 · · · a0 0 · · · · · · · · ·a(n/2)+2 a(n/2)+1 a(n/2) + U2 a(n/2)−1 · · · a0 0 · · · · · ·

· · · · · · · · · · · · · · · · · · · · · 0 · · · · · ·an an−1 · · · a(n/2)+1 a(n/2) + U(n/2) a(n/2)−1 · · · a0 0 · · ·0 an an−1 · · · a(n/2)+1 a(n/2) + U(n/2)+1 a(n/2)−1 · · · a0 0 · · ·0 0 an an−1 · · · a(n/2)+1 a(n/2) + U(n/2)+2 · · · · · · a0 · · ·· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

(22)

E =

E0 0 0 · · · 00 E1 0 · · · 00 0 E2 · · · 0· · · · · · · · · · · · · · ·0 0 0 · · · EN

(23)

and

Y = (y0, y1, y2, y3, . . . , yN)t (24)

where (N +1) is the total number of discretized points and a superscript t denotes the transposeof a vector. Though there are (N + 1) eigenvalues and eigenfunctions, only a small numberof states that have physical meaning are required. We have to appropriately choose the wholeinterval in order to avoid deterioration in the accuracy of the eigenvalues and eigenfunctions.The whole interval should be selected so that the magnitude of the tail of the eigenfunction atboth ends is small enough, but not too small, in order to avoid deterioration due to numericalerrors. From our numerical experiment, the whole interval is adjusted so that the magnitude ofthe tail of the normalized eigenfunction with the maximum quantum number that is required bythe problem ranges over 1.0D-15 to 1.0D-10 at both ends of the whole interval in order to get therelative errors of the eigenvalues around 1.0D-15. Though the formula of the lowest order forthe second derivative, [(d/dx)2y]i = (yi+1−2yi +yi−1)/h2, has been extensively used [48–50],its accuracy is not good enough. By using a high degree formula for the second derivative andchoosing the appropriate whole interval, we obtain accurate eigenvalues and eigenfunctionsas shown later, though the accuracy was not good enough in [23–25]. The necessary matrix


eigenvalue solvers have been provided by many authors [35–37]. The eigenvalue is calculatedby using the Householder transformation for matrix tridiagonalization and the bisection methodbased on Sturm’s theorem [35,36]. The eigenvector is calculated by using an inverse iterationmethod [35, 36] and normalized by using the accurate central-difference integration formulain section 3.3.

Equation (4) may be regarded as a self-consistent equation for Eν because theeigenfunction is calculated by using the eigenvalue. Now we can calculate the matrix elementson the right-hand side by using the accurate numerical differentiation and integration describedin the preceding subsections. Both the eigenvalues and matrix elements converge to their exactvalues by increasing the degree n of the second derivative. Numerical experience indicates thatthe matrix elements converge faster than the eigenvalues, as will be shown later. Coincidence ofthe two computed quantities provides a method of cross-checking the accuracy between them.

In concluding this section we note the relative and absolute errors. We show the relativeerror or accurate digits for eigenvalues and integrals such as matrix elements. We showabsolute errors for integrands such as normalized wavefunctions and their derivatives, sinceabsolute-error control is important for such integrands for practical numerical integration.

4. Results and discussion

4.1. Linear harmonic oscillator

As a typical application of the new calculation method, we take the linear harmonic oscillator,because the analytic properties of its solution are well known. The potential is U(x) =12mω2x2, where ω is the angular frequency [1–6]. The potential is symmetric with respect tox = 0 and infinite as |x| → ∞. The number of bound states is infinite for this potential. TheSchrodinger equation can be reduced to dimensionless form by introducing units of energyE0 = 1

2 hω and of length ξ = αx, α = (h/mω)1/2 and λ = E/E0, so that it can be rewrittenas

(−d2/dξ 2 + ξ 2)ψ = λψ. (25)

The solution with quantum number ν is given by the Hermite polynomial Hν(ξ):

λν = 2ν + 1, ν = 0, 1, 2, . . . (26)

ψν(ξ) = (α/π1/22νν!)1/2 exp[− 12ξ 2]Hν(ξ). (27)

The discrete mesh points along the ξ axis are allocated with equidistant width h. Thewavefunctions are tabulated at these points.

Figure 1(a) shows the magnitude of the absolute errors of the wavefunctions for ν = 0–7in the interval [0, 10], evaluated at the centres of the interval h = 1

64 by using the Lagrangeinterpolation. The absolute errors of interpolation with degree 9 are less than 1.0D-15 andthe interpolation often gives exact values shown by nearly vertical lines. Interpolation withdegree 3 has absolute errors of less than 1.0D-7. The errors as a function of ξ and ν canbe understood in terms of the derivative f (n+1)(ξ) in equation (8). Applying the well-knownrecurrence relation (d/dξ)ψν(ξ) = (ν/2)1/2ψν−1(ξ) − [(ν + 1)/2]1/2ψν+1(ξ) n times, we seethat (d/dξ)n+1ψν(ξ) is given by a product of exp[− 1

2ξ 2] and a sum of the Hermite polynomialswith maximum degree ξν+n+1 of Hν+n+1(ξ). The asymptotic form of the error at large ξ isdetermined by exp[− 1

2ξ 2] and the error increases for large ν because of the polynomial withmaximum degree ξν+n+1. Figure 1(b) shows the absolute error as a function of degree forξ = 0.492 1875 with ν = 0 for h = 1

64 , 132 , 1

16 and 18 . With increasing degree, the error

decreases for all ξ and ν, rapidly for smaller h, and is below 1.0D-15 for degree 9 at h = 164

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0.0010.1

0 2 4 6 8 10

ν = 0 ν = 1 ν = 2 ν = 3 ν = 4 ν = 5 ν = 6 ν = 7 1D-15 ν = 0 ν = 1 ν = 2 ν = 3 ν = 4 ν = 5 ν = 6 ν = 7

Degree 3

Degree 9A

bso

lute

err

ors

of

inte

rpo

lati

on

X

(a)

10-18

10-16

10-14

10-12

10-10

10-8

10-6

2 4 6 8 10 12

h = 1/64 h = 1/32 h = 1/16 h = 1/8 1D-15

ν = 0x = 0.4921875

Degree

Ab

solu

te e

rro

rs o

f in

terp

ola

tio

nA

bso

lute

err

ors

of

inte

rpo

lati

on

(b)

Figure 1. Absolute errors of interpolation for (a) wavefunctions with quantum numbers ν = 0–7at centres of the interval h using Lagrange interpolation of degree 3 and degree 9, and for (b)wavefunctions with ν = 0 as a function of the degree of Lagrange interpolation for h = 1

64 , 132 , 1

16

and 18 . Absolute values are taken for the errors.

and 132 . The errors as a function of n and h can be understood in terms of πn(ξ)/(n + 1)!

in equation (8); they are decreasing functions of n with a factor hn. These results show thatthe polynomial interpolation provides simple and accurate evaluation of functions for mostpractical purposes.

Figure 2(a) shows absolute errors in the first derivative of the wavefunctions for ν = 0–7,numerically evaluated at the mesh points for the interval h = 1

64 . The absolute errors in thefirst derivative with degree 10 are less than 1.0D-14 and the first derivative also gives exactvalues shown by nearly vertical lines. The first derivative with degree 4 has absolute errorsof less than 1.0D-6. The errors as a function of ξ and ν can be understood in terms of thef (n+1)(ξ) of the truncation error cn,1h

nf (n+1)(ξ). Figure 2(b) shows the absolute error as afunction of the degree for ξ = 0.5 with ν = 1 for h = 1

64 , 132 , 1

16 and 18 . The error decreases

with increasing degree for smaller h. The flattening of the error for small h = 164 at n larger


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0.0010.1

0 2 4 6 8 10


Degree 10

Degree 4

(a)

Ab

solu

te e

rro

rs in

1st

der

ivat

ive

X

10-18

10-16

10-14

10-12

10-10

10-8

10-6

0.0001

0.01

2 4 6 8 10

h = 1/64 h = 1/32 h = 1/16 h = 1/8 1D-15

ν = 1x = 0.5

Ab

solu

te e

rro

rs in

1st

der

ivat

ive

Degree

(b)

Figure 2. Absolute errors in numerical first derivative of (a) wavefunctions for quantum numbersν = 0–7 at mesh points with the interval h using first derivative formulae of degree 4 and degree 10,and of (b) wavefunctions with ν = 1 as a function of the degree of first derivative formulae forh = 1

64 , 132 , 1

16 and 18 . Absolute values are taken for the errors.

than 8 is due to the round-off error proportional to 1/h for the first derivative. These resultsshow that the numerical first derivative provides accurate evaluation of the derivatives for mostpractical purposes.

Figure 3(a) shows absolute errors in the second derivative of the wavefunctions for ν = 0–7, numerically evaluated at the mesh points for the interval h = 1

64 using equation (10) withm = 2. The absolute errors in the second derivative with degree 10 are less than 1.0D-12and those with degree 4 are less than 1.0D-6. The errors as a function of ξ and ν can also beunderstood in terms of the truncation error cn,2h

nf (n+2)(ξ). Figure 3(b) shows the absoluteerror as a function of the degree for ξ = 0.5 with ν = 1 for h = 1

64 , 132 , 1

16 and 18 evaluated by

using equations (10)–(12) with m = 2. For the two larger values of h (h = 18 and 1

16 ), the errordecreases monotonically or becomes constant with increasing degree. However, for the two

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0 2 4 6 8 10


Degree 10

Degree 4

(a)

X

Ab

solu

te e

rro

rs in

2n

d d

eriv

ativ

e

10-18

10-16

10-14

10-12

10-10

10-8

10-6

0.0001

0.01

2 4 6 8 10 12 14

h = 1/64 h = 1/32 h = 1/16 h = 1/8 1D-15

ν = 1x = 0.5

(b)

DegreeDegreeDegree

Ab

solu

te e

rro

rs in

2n

d d

eriv

ativ

e

Figure 3. Absolute errors in numerical second derivative of (a) wavefunctions for quantum numbersν = 0–7 at mesh points with the interval h using second derivative formulae of degree 4 anddegree 10, and of (b) wavefunctions with ν = 1 as a function of the degree of second derivativeformulae for h = 1

64 , 132 , 1

16 and 18 . Absolute values are taken for the errors.

smaller values of h (h = 132 and 1

64 ), the error shows non-monotonic behaviour for degreesfrom 8 to 12 but is monotonic above and below these values. The non-monotonic behaviourthat is conspicuous for smaller h is due to the round-off error proportional to 1/h2 for thesecond derivative. The numerical second derivative also provides accurate evaluation of thederivatives for most practical purposes.

The performance of integration and calculation of the integrand are clearly seen bychecking the accuracy of the orthonormal integrals for the wavefunctions of the linear harmonicoscillator shown in table 1. The second column shows the orthonormal integrals obtained byusing the central-difference integration formula for which the function is calculated exactlyonly at the mesh points. The numerical result demonstrates 15-digit accuracy of the central-difference integration formula with degree 8 and h = 1

64 . The third column shows numerical


Table 1. Orthonormal integrals 〈ν|ν′〉 of wavefunctions with quantum numbers ν and ν′ of a linearharmonic oscillator. CDIF: central-difference integration formula with degree 8. GHE: Gauss–Hermite quadrature rule with exact integrand at 14 abscissas. GHI: Gauss–Hermite quadrature rulewith interpolated integrand at 14 abscissas. MNIPGH: minimum number of integration points forthe Gauss–Hermite quadrature rule for 15-digit accuracy.

〈ν|ν′〉 CDIF GHE GHI MNIPGH

〈0|0〉 1.000 000 000 000 000 1.000 000 000 000 000 1.000 000 000 000 000 1〈1|1〉 1.000 000 000 000 000 0.999 999 999 999 999 1.000 000 000 000 000 2〈2|2〉 1.000 000 000 000 000 1.000 000 000 000 000 1.000 000 000 000 000 3〈3|3〉 1.000 000 000 000 000 1.000 000 000 000 000 1.000 000 000 000 000 4〈4|4〉 0.999 999 999 999 999 0.999 999 999 999 999 1.000 000 000 000 000 5〈5|5〉 1.000 000 000 000 000 1.000 000 000 000 000 1.000 000 000 000 000 6〈6|6〉 1.000 000 000 000 000 1.000 000 000 000 000 1.000 000 000 000 000 6〈7|7〉 1.000 000 000 000 000 1.000 000 000 000 000 1.000 000 000 000 000 7〈8|8〉 0.999 999 999 999 999 1.000 000 000 000 000 1.000 000 000 000 000 8〈9|9〉 0.999 999 999 999 999 0.999 999 999 999 998 1.000 000 000 000 000 9Others 0.000 000 000 000 000 0.000 000 000 000 000 0.000 000 000 000 000 2–12

Table 2. Matrix elements 〈ν|ξ |ν′〉 for wavefunctions with quantum numbers ν and ν′ of a linearharmonic oscillator by using the central-difference integration formula (CDIF).

〈ν|ξ |ν′〉 CDIF Exact

〈0|ξ |1〉 7.071 067 811 865 47D−01 0.51/2 = 7.071 067 811 865 48D−01〈1|ξ |2〉 9.999 999 999 999 99D−01 1.0〈2|ξ |3〉 1.224 744 871 391 59D+00 1.51/2 = 1.224 744 871 391 59D+00〈3|ξ |4〉 1.414 213 562 373 09D+00 2.01/2 = 1.414 213 562 373 10D+00〈4|ξ |5〉 1.581 138 830 084 19D+00 2.51/2 = 1.581 138 830 084 19D+00〈5|ξ |6〉 1.732 050 807 568 88D+00 3.01/2 = 1.732 050 807 568 88D+00〈6|ξ |7〉 1.870 828 693 386 97D+00 3.51/2 = 1.870 828 693 386 97D+00〈7|ξ |8〉 2.000 000 000 000 00D+00 2.0〈8|ξ |9〉 2.121 320 343 559 64D+00 4.51/2 = 2.121 320 343 559 64D+00〈9|ξ |10〉 2.236 067 977 499 79D+00 5.01/2 = 2.236 067 977 499 79D+00Others 0.000 000 000 000 00 0

integration using the Gauss–Hermite quadrature rule with exactly evaluated functions at 14abscissas. The 15-digit accuracy is clearly seen. The fourth column shows numericalintegration by using the Gauss–Hermite quadrature rule with wavefunctions evaluated usingpolynomial interpolation at the abscissas. We obtained 15-digit accuracy, which also confirmedthe interpolation with 15-digit accuracy. The fifth column shows the minimum number ofintegration points for the Gauss–Hermite quadrature rule for 15-digit accuracy. In addition tothese, we evaluated the orthonormal integrals up to ν = 32 which also show the same accuracy,but these are omitted for brevity.

The matrix elements for coordinate 〈ν|ξk|ν ′〉, k = 1, 2, are shown in tables 2 and 3,respectively. The central-difference integration formula in this case also gives accurateintegration as for the orthonormal integrals. The numerical integration provides both diagonaland off-diagonal matrix elements with the same accuracy, in contrast to the method that providesonly diagonal matrix elements [12–20]. Further, we evaluated the matrix elements up to ν = 32which show the same accuracy, but are again omitted for brevity. In addition to these we alsocalculated matrix elements (not shown for brevity) for ξ 3 and ξ 4 which show the same highperformance as for ξ and ξ 2.

4464 H Ishikawa

Table 3. Matrix elements 〈ν|ξ2|ν′〉 for wavefunctions with quantum numbers ν and ν′ of a linearharmonic oscillator by using the central-difference integration formula (CDIF).

〈ν|ξ2|ν′〉 CDIF Exact

〈0|ξ2|0〉 5.000 000 000 000 00D−01 0.5〈0|ξ2|2〉 7.071 067 811 865 47D−01 0.51/2 = 7.071 067 811 865 48D−01〈1|ξ2|1〉 1.500 000 000 000 00D+00 1.5〈1|ξ2|3〉 1.224 744 871 391 59D+00 1.51/2 = 1.224 744 871 391 59D+00〈2|ξ2|2〉 2.500 000 000 000 00D+00 2.5〈2|ξ2|4〉 1.732 050 807 568 88D+00 3.01/2 = 1.732 050 807 568 88D+00〈3|ξ2|3〉 3.500 000 000 000 00D+00 3.5〈3|ξ2|5〉 2.236 067 977 499 79D+00 5.01/2 = 2.236 067 977 499 79D+00〈4|ξ2|4〉 4.500 000 000 000 00D+00 4.5〈4|ξ2|6〉 2.738 612 787 525 83D+00 7.51/2 = 2.738 612 787 525 83D+00〈5|ξ2|5〉 5.500 000 000 000 00D+00 5.5〈5|ξ2|7〉 3.240 370 349 203 93D+00 10.51/2 = 3.240 370 349 203 93D+00〈6|ξ2|6〉 6.500 000 000 000 00D+00 6.5〈6|ξ2|8〉 3.741 657 386 773 94D+00 14.01/2 = 3.741 657 386 773 94D+00〈7|ξ2|7〉 7.500 000 000 000 00D+00 7.5〈7|ξ2|9〉 4.242 640 687 119 29D+00 18.01/2 = 4.242 640 687 119 28D+00〈8|ξ2|8〉 8.500 000 000 000 00D+00 8.5〈8|ξ2|10〉 4.743 416 490 252 57D+00 22.51/2 = 4.743 416 490 252 57D+00〈9|ξ2|9〉 9.500 000 000 000 00D+00 9.5〈9|ξ2|11〉 5.244 044 240 850 76D+00 27.51/2 = 5.244 044 240 850 76D+00Others 0.000 000 000 000 00 0

Table 4. Matrix elements 〈ν|d/dξ |ν′〉 for wavefunctions with quantum numbers ν and ν′ of a linearharmonic oscillator by using the central-difference integration formula.

〈ν|d/dξ |ν′〉 Numerical derivative Analytic derivative Exact

〈0|d/dξ |1〉 7.071 067 811 865 47D−01 7.071 067 811 865 47D−01 0.51/2 = 7.071 067 811 865 48D−01〈1|d/dξ |2〉 1.000 000 000 000 00D+00 1.000 000 000 000 00D+00 1.0〈2|d/dξ |3〉 1.224 744 871 391 59D+00 1.224 744 871 391 59D+00 1.51/2 = 1.224 744 871 391 59D+00〈3|d/dξ |4〉 1.414 213 562 373 10D+00 1.414 213 562 373 09D+00 2.01/2 = 1.414 213 562 373 10D+00〈4|d/dξ |5〉 1.581 138 830 084 19D+00 1.581 138 830 084 19D+00 2.51/2 = 1.581 138 830 084 19D+00〈5|d/dξ |6〉 1.732 050 807 568 88D+00 1.732 050 807 568 88D+00 3.01/2 = 1.732 050 807 568 88D+00〈6|d/dξ |7〉 1.870 828 693 386 97D+00 1.870 828 693 386 97D+00 3.51/2 = 1.870 828 693 386 97D+00〈7|d/dξ |8〉 2.000 000 000 000 00D+00 2.000 000 000 000 00D+00 2.0〈8|d/dξ |9〉 2.121 320 343 559 64D+00 2.121 320 343 559 64D+00 4.51/2 = 2.121 320 343 559 64D+00〈9|d/dξ |10〉 2.236 067 977 499 79D+00 2.236 067 977 499 79D+00 5.01/2 = 2.236 067 977 499 79D+00Others 0.000 000 000 000 00 0.000 000 000 000 00 0

We show in table 4 the matrix elements for the derivative 〈ν|d/dξ |ν ′〉 by using thecentral-difference integration formula. The third column, calculated with the analyticsolution of the derivatives, indicates the accuracy of the central-difference integration formulain comparison with the exact results shown in the fourth column. The second column showsthat the numerical differentiation at the mesh points is accurate.

The matrix elements for 〈ν|ξ |ν ′〉 and 〈ν|d/dξ |ν ′〉 are the dipole and momentum matrixelements, respectively, in the optical transitions in quantum mechanics and the equality〈ν|ξ |ν ′〉 = 〈ν|d/dξ |ν ′〉 holds to 15-digit accuracy according to tables 2 and 4.


10-17

10-15

10-13

10-11

10-9

10-7

10-5

0.001

2 4 6 8 10 12

ν = 0 (MD) ν = 0 (ME) ν = 3 (MD) ν = 3 (ME) ν = 6 (MD) ν = 6 (ME) ν = 9 (MD) ν = 9 (ME) 1D-15

Rel

ativ

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(a)

10-17

10-15

10-13

10-11

10-9

10-7

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2 4 6 8 10 12

ν = 0 ν = 3 ν = 6 ν = 9 1D-15

Ab

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rs in

eig

enfu

nct

ion

Degree

(b)

Figure 4. (a) Relative errors in eigenvalues for quantum numbers ν = 0–9 as a function of degreen of the second derivative formulae and relative errors in the matrix elements Eν . MD indicatesthe method of discretized matrix equation and ME denotes matrix elements. (b) Absolute errors inwavefunctions for ν = 0–9 as a function of degree n of the second derivative formulae. Absolutevalues are taken for the errors.

We show in table 5 the matrix elements for the second derivative 〈ν|(d/dξ)2|ν ′〉. Thesecond column, calculated using the numerical derivatives, where the second derivative iscalculated by using the first derivative twice, indicates the accuracy of the numerical derivative.The third column shows that the formula for the analytic second derivative at the mesh pointscoincides with the exact result. We also calculated the second derivative by using equation (10)with m = 2, which shows similar results.

The diagonal matrix elements for 〈T 〉 = 〈ν| − (d/dξ)2|ν〉 and 〈V 〉 = 〈ν|ξ 2|ν〉 arekinetic and potential energies, respectively, and the relation 〈T 〉 = 〈V 〉 = 1

2Eν , Eν =〈ν| − (d/dξ)2 + ξ 2|ν〉/〈ν|ν〉, for the linear harmonic oscillator holds to 15-digit accuracyas shown in tables 3, 5 and 6.

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Table 5. Matrix elements 〈ν|(d/dξ)2|ν′〉 for wavefunctions with quantum numbers ν and ν′ of a linear harmonic oscillator by using the central-differenceintegration formula.

〈ν|(d/dξ)2|ν′〉 Numerical derivative Analytic derivative Exact

〈0|(d/dξ)2|0〉 −5.000 000 000 000 00D−01 −5.000 000 000 000 00D−01 −0.5〈0|(d/dξ)2|2〉 7.071 067 811 865 48D−01 7.071 067 811 865 48D−01 0.51/2 = 7.071 067 811 865 48D−01〈1|(d/dξ)2|1〉 −1.500 000 000 000 00D+00 −1.500 000 000 000 00D+00 −1.5〈1|(d/dξ)2|3〉 1.224 744 871 391 59D+00 1.224 744 871 391 59D+00 1.51/2 = 1.224 744 871 391 59D+00〈2|(d/dξ)2|2〉 −2.500 000 000 000 00D+00 −2.500 000 000 000 00D+00 −2.5〈2|(d/dξ)2|4〉 1.732 050 807 568 88D+00 1.732 050 807 568 88D+00 3.01/2 = 1.732 050 807 568 88D+00〈3|(d/dξ)2|3〉 −3.500 000 000 000 00D+00 −3.500 000 000 000 00D+00 −3.5〈3|(d/dξ)2|5〉 2.236 067 977 499 79D+00 2.236 067 977 499 79D+00 5.01/2 = 2.236 067 977 499 79D+00〈4|(d/dξ)2|4〉 −4.500 000 000 000 00D+00 −4.500 000 000 000 00D+00 −4.5〈4|(d/dξ)2|6〉 2.738 612 787 525 83D+00 2.738 612 787 525 83D+00 7.51/2 = 2.738 612 787 525 83D+00〈5|(d/dξ)2|5〉 −5.500 000 000 000 00D+00 −5.500 000 000 000 00D+00 −5.5〈5|(d/dξ)2|7〉 3.240 370 349 203 93D+00 3.240 370 349 203 93D+00 10.51/2 = 3.240 370 349 203 93D+00〈6|(d/dξ)2|6〉 −6.500 000 000 000 00D+00 −6.500 000 000 000 00D+00 −6.5〈6|(d/dξ)2|8〉 3.741 657 386 773 94D+00 3.741 657 386 773 94D+00 14.01/2 = 3.741 657 386 773 94D+00〈7|(d/dξ)2|7〉 −7.500 000 000 000 00D+00 −7.500 000 000 000 00D+00 −7.5〈7|(d/dξ)2|9〉 4.242 640 687 119 28D+00 4.242 640 687 119 28D+00 18.01/2 = 4.242 640 687 119 28D+00〈8|(d/dξ)2|8〉 −8.499 999 999 999 99D+00 −8.500 000 000 000 00D+00 −8.5〈8|(d/dξ)2|10〉 4.743 416 490 252 56D+00 4.743 416 490 252 57D+00 22.51/2 = 4.743 416 490 252 57D+00〈9|(d/dξ)2|9〉 −9.500 000 000 000 00D+00 −9.500 000 000 000 00D+00 −9.5〈9|(d/dξ)2|11〉 5.244 044 240 850 75D+00 5.244 044 240 850 76D+00 27.51/2 = 5.244 044 240 850 76D+00Others 0.000 000 000 000 000 0.000 000 000 000 000 0


Table 6. Matrix elements Eν = 〈ν|H |ν〉/〈ν|ν〉 = 〈ν|−(d/dξ)2 +U(ξ)|ν〉/〈ν|ν〉 for wavefunctionswith quantum number ν of a linear harmonic oscillator by using the central-difference integrationformula.

Quantum no Numerical derivative Analytic derivative Exact

0 1.000 000 000 000 00D+00 1.000 000 000 000 00D+00 1.01 3.000 000 000 000 00D+00 3.000 000 000 000 00D+00 3.02 5.000 000 000 000 00D+00 5.000 000 000 000 00D+00 5.03 7.000 000 000 000 00D+00 7.000 000 000 000 00D+00 7.04 9.000 000 000 000 00D+00 9.000 000 000 000 00D+00 9.05 1.100 000 000 000 00D+01 1.100 000 000 000 00D+01 11.06 1.300 000 000 000 00D+01 1.300 000 000 000 00D+01 13.07 1.500 000 000 000 00D+01 1.500 000 000 000 00D+01 15.08 1.700 000 000 000 00D+01 1.700 000 000 000 00D+01 17.09 1.900 000 000 000 00D+01 1.900 000 000 000 00D+01 19.0

Table 7. Eigenvalues of the discretized matrix equation and matrix elements Eν = 〈ν|− (d/dξ)2 +U(ξ)|ν〉/〈ν|ν〉 for a wavefunction with quantum number ν of a linear harmonic oscillator by usingthe central-difference integration formula. The fifth column is the maximum of the absolute valueof the absolute error for the wavefunctions (MAVAEWF).

Quantum no Eigenvalue Eν Exact MAVAEWF

0 1.000 000 000 000 25D+00 1.000 000 000 000 00D+00 1.0 1.512D−131 3.000 000 000 000 27D+00 3.000 000 000 000 00D+00 3.0 2.476D−132 4.999 999 999 999 82D+00 4.999 999 999 999 99D+00 5.0 3.552D−133 7.000 000 000 000 25D+00 6.999 999 999 999 96D+00 7.0 3.413D−134 9.000 000 000 000 25D+00 8.999 999 999 999 87D+00 9.0 4.165D−135 1.100 000 000 000 02D+01 1.099 999 999 999 96D+01 11.0 2.358D−136 1.300 000 000 000 02D+01 1.299 999 999 999 91D+01 13.0 2.212D−137 1.499 999 999 999 93D+01 1.499 999 999 999 80D+01 15.0 3.189D−138 1.699 999 999 999 98D+01 1.699 999 999 999 59D+01 17.0 4.456D−139 1.900 000 000 000 04D+01 1.899 999 999 999 22D+01 19.0 2.548D−13

The result for the discretized matrix equation method for the linear harmonic oscillator,equations (21)–(24) and (4), is shown in figure 4. The relative errors of the eigenvaluedecrease as a function of the degree n and converge to less than 5.0D-13 for n larger thanor equal to 12 (figure 4(a)). The relative error increases for large ν where the error is largerthan 5.0D-13. The relative errors of the matrix elements Eν are smaller and decrease faster thanthose of the eigenvalues for errors larger than 5.0D-13 and converge at degree 6. These resultsindicate that the relative errors are less than 5.0D-13, substantially less in fact. The maximumabsolute errors in the normalized eigenfunctions decrease monotonically as a function ofthe degree and converge below 5.0D-13 for degree n larger than or equal to 8, as shown infigure 4(b). Table 7 summarizes the results for degree 12, the whole interval (−10.0, 10.0)

and h = 132 . The accuracy of the eigenvalue in the second column is from 13 to 14 digits

and that of the matrix element Eν in the third column ranges from 13 to 15 digits and is ofthe same order as for the eigenvalue. The magnitude of the absolute errors for the normalizedwavefunctions is less than 5.0D-13, as seen in the fifth column. In [51] the discretized matrixmethod of Numerov with defect correction was used for calculating eigenvalues of the linearharmonic oscillator and the relative error of the eigenvalues ranged widely from 1.3D-14 forν = 0 to 2.4D-12 for ν = 3. In concluding this subsection it is noted again that analysis ofthe convergence for discretized matrix eigenvalues and matrix elements of the Hamiltonianprovides a good cross-check for the eigenvalues.

4468 H Ishikawa

4.2. Anharmonic oscillator

The second example is an anharmonic oscillator of the form U(ξ) = µξ 2 + λξ 4, where µ

and λ are constants [12, 14, 15, 18, 19, 23, 24, 52–70]. The potential has a single minimum forµ � 0 but a double minimum for µ < 0, symmetric with respect to ξ = 0 and the potentialbecomes infinite as |ξ | → ∞ for positive λ. The number of bound states is infinite for thispotential. The eigenvalue for the bound state has been frequently investigated for a wide rangeof physics applications and accurate eigenvalues were numerically obtained by using othermethods of solution [52–70]. In order to illustrate the performance of our simple method, wetake three typical cases for (µ, λ) = (0.0, 1.0), (1.0, 1.0) and (−1.0, 1.0) in table 8, where thedegree and the whole interval are 12 and (−4.843 75, 4.843 75) for the first two cases, and 14and (−4.531 25, 4.531 25) for the last one. The eigenvalues of the matrix equation and matrixelements Eν are of 13-digit to 15-digit accuracy, which is much higher than the 7-digit accuracyobtained by using the discretized matrix equation with low degree [23, 24]. The accuracy ofthe present work is comparable to the best values ever reported [58,60,62,66,68,69] in double-precision arithmetic operations. For the double-minimum case (µ, λ) = (−1.0, 1.0) we showten states that probably have 13-digit or higher accuracy, though the results using other methodshave not been shown.

The third example is an anharmonic oscillator of the form U(ξ) = ξ 2 + λξ 2/(1 + gξ 2),where λ and g are constants [18,23,24,71–79] in the reduced units of E0 = h2/(2mα2) and oflength ξ = αx. The potential is symmetric with respect to ξ = 0 and infinite as |ξ | → ∞. Thenumber of bound states is also infinite for this potential. The eigenvalue for the bound statehas been investigated in detail and exact eigenvalues were obtained for special combinationsof λ, g and the quantum number ν [18,23,24,71–79]. We take four typical cases for (λ, g, ν),indicated by the notation (∗) in table 9. The eigenvalues of the matrix are of 13-digit accuracyand Eν are of 15-digit accuracy for degree 12, the whole interval (−10.0, 10.0) and h = 1

32 ;the accuracy is comparable to the best values ever reported [23, 75–79] in double-precisionarithmetic operations. For another typical case, λ = g = 1.0, we show ten states whoseeigenvalues and Eν coincide with those in [78, 79] within the accuracy referred to therein.

4.3. Morse potential and modified Poschl–Teller potential

The fourth example is the nonlinear Morse potential U(x) = V0(e−2αx − 2e−αx) [3–5, 80].The potential is non-symmetric with respect to x = 0 and has a finite range between −V0

and zero for x > 0 and is infinite as x → −∞. The number of bound states is finite for thispotential. The Schrodinger equation can also be reduced to dimensionless form by introducingunits of energy E0 = h2/(2mα2) and of length ξ = αx. The eigenvalue for the bound statewith quantum number ν is given by [3–5]

λν = (Eν/E0) = −(V0/E0)[1 − (ν + 0.5)/(V0/E0)1/2]2 (28)

where ν = 0, 1, 2, . . . with ν < (V0/E0)1/2 − 0.5. We show typical cases for V0/E0 in

table 10, where the degree is 14 and h = 132 , the whole interval is (−4.1875, 35.8125)

for V0/E0 = 1.0, (−3.781 25, 27.468 75) for 2.25, (−3.281 25, 21.718 75) for 6.25 and(−2.96, 22.64) for 12.25. The eigenvalues and the matrix elements of the Hamiltonian are of13- to 15-digit accuracy.

Another form of the Morse potential [2, 6, 25, 51, 81–84] is U(x) = D{1 − exp[−α(x −x0)]}2 with D = ω2

e/4ωexe, α = (kωexe)1/2 and k = 1, having the theoretical eigenvalues

Eν = ωe(ν + 12 ) − ωexe(ν + 1

2 )2. (29)

The eigenvalues and matrix elements of the Hamiltonian for the case x0 = 2.408 73,ωe = 48.668 88 and ωexe = 0.977 888 [25, 84], are of 13- to 15-digit accuracy for the degree

An

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4469

Table 8. Eigenvalues of the discretized matrix equation and matrix elements Eν = 〈ν| − (d/dξ)2 + U(ξ)|ν〉/〈ν|ν〉 for a wavefunction with quantumnumber ν of a potential V (ξ) = µξ2 + λξ4 by using the central-difference integration formula.

µ λ Quantum no Eigenvalue Eν Other methods

0.0 1.0 0 1.060 362 090 484 94D+00 1.060 362 090 484 18D+00 1.060 362 090 484 18D+00a,b,c,d

1 3.799 673 029 799 24D+00 3.799 673 0298 014 0D+00 3.799 673 029 801 40D+00a,b,c

2 7.455 697 937 987 76D+00 7.455 697 937 986 74D+00 7.455 697 937 986 74D+00a,b

3 1.164 474 551 137 86D+01 1.164 474 551 137 82D+01 1.164 474 551 137 82D+01a,b

4 1.626 182 601 885 11D+01 1.626 182 601 885 02D+01 1.626 182 601 885 02D+01a

5 2.123 837 291 823 54D+01 2.123 837 291 823 59D+01 2.123 837 291 823 60D+01a

6 2.652 847 118 368 03D+01 2.652 847 118 368 24D+01 2.652 847 118 368 25D+01a

7 3.209 859 771 096 60D+01 3.209 859 771 096 80D+01 3.209 859 771 096 83D+01a

8 3.792 300 102 703 42D+01 3.792 300 102 703 30D+01 3.792 300 102 703 40D+01a

9 4.398 115 809 729 02D+01 4.398 115 809 728 74D+01 4.398 115 809 728 97D+01a

1.0 1.0 0 1.392 351 641 520 79D+00 1.392 351 641 530 29D+00 1.392 351 641 530 29D+00a,b,c,e

1 4.648 812 704 209 47D+00 4.648 812 704 212 08D+00 4.648 812 704 212 08D+00a,b,c,e

2 8.655 049 957 754 54D+00 8.655 049 957 759 32D+00 8.655 049 957 759 31D+00a,b,e

3 1.315 680 389 804 50D+01 1.315 680 389 804 99D+01 1.315 680 389 804 99D+01a,b,e

4 1.805 755 743 630 02D+01 1.805 755 743 630 32D+01 1.805 755 743 630 33D+01a,e

5 2.329 744 145 121 87D+01 2.329 744 145 122 31D+01 2.329 744 145 122 32D+01a,e

6 2.883 533 845 950 03D+01 2.883 533 845 950 41D+01 2.883 533 845 950 42D+01a,e

7 3.464 084 832 110 68D+01 3.464 084 832 111 08D+01 3.464 084 832 111 13D+01a,e

8 4.069 038 608 210 14D+01 4.069 038 608 210 51D+01 4.069 038 608 210 64D+01a,e

9 4.696 500 950 567 40D+01 4.696 500 950 567 24D+01 4.696 500 950 567 55D+01a,e

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Table 8. (Continued.)

µ λ Quantum no Eigenvalue Eν Other methods

−1.0 1.0 0 6.576 530 051 780 60D−01 6.576 530 051 807 15D−01 6.576 530 051 807 15D−01b,f

1 2.834 536 202 116 43D+00 2.834 536 202 119 30D+00 2.834 536 202 119 30D+00b,f

2 6.163 901 256 958 67D+00 6.163 901 256 963 07D+003 1.003 864 612 070 77D+01 1.003 864 612 071 16D+014 1.437 240 650 467 45D+01 1.437 240 650 467 79D+015 1.908 571 468 502 16D+01 1.908 571 468 502 42D+016 2.412 807 549 278 05D+01 2.412 807 549 278 22D+017 2.946 285 591 419 97D+01 2.946 285 591 420 11D+018 3.506 214 903 107 54D+01 3.506 214 903 107 60D+019 4.090 385 627 182 25D+01 4.090 385 627 182 30D+01

a Banerjee et al [60].b Fernandez et al [69].c Schiffrer and Stanzial [68].d Biswas et al [58].e Banerjee [62].f Basla et al [66].

An

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4471

Table 9. Eigenvalues of the discretized matrix equation and matrix elements Eν = 〈ν| − (d/dξ)2 + U(ξ)|ν〉/〈ν|ν〉 for a wavefunction with quantumnumber ν of a potential V (ξ) = ξ2 + λξ2/(1 + gξ2) by using the central-difference integration formula.

λ g Quantum no Eigenvalue Eν Other methods

−0.42 0.1 0 8.000 000 000 004 12D−01 8.000 000 000 000 00D−01 0.8∗,a

1 2.455 698 585 118 43D+00 2.455 698 585 119 10D+00 2.455 698 585 119b

2 4.197 895 893 444 87D+00 4.197 895 893 444 28D+00 4.197 895 893 444b

3 5.991 398 837 190 70D+00 5.991 398 837 189 80D+00 5.991 398 837 190b

4 7.820 097 654 268 70D+00 7.820 097 654 268 44D+00 7.820 097 654 268b

5 9.674 537 312 905 86D+00 9.674 537 312 906 14D+00 9.674 537 312 906b

−0.46 0.1 1 2.399 999 999 999 43D+00 2.400 000 000 000 00D+00 2.4∗,c

−0.495 357 508 034 270 0.1 2 4.046 424 919 656 84D+00 4.046 424 919 657 30D+00 4.046 424 919 657 30∗,d

−0.527 762 515 838 433 0.1 3 5.722 374 841 615 66D+00 5.722 374 841 615 66D+00 5.722 374 841 615 67∗,e

1.0 1.0 0 1.232 350 723 405 27D+00 1.232 350 723 406 06D+00 1.232 350 723 406 06f

1 3.507 388 348 905 61D+00 3.507 388 348 905 28D+00 3.507 388 348 905b

2 5.589 778 933 736 18D+00 5.589 778 933 737 15D+00 5.589 778 933 736b

3 7.648 201 241 718 89D+00 7.648 201 241 719 34D+00 7.648 201 241 723b

4 9.684 042 015 229 18D+00 9.684 042 015 229 99D+00 9.684 042 015 230 17f

5 1.171 223 747 020 79D+01 1.171 223 747 020 79D+016 1.373 324 101 210 73D+01 1.373 324 101 210 84D+017 1.575 063 879 714 55D+01 1.575 063 879 714 41D+018 1.776 477 910 142 13D+01 1.776 477 910 141 69D+019 1.977 689 487 169 51D+01 1.977 689 487 168 65D+01

∗ Exact eigenvalue.a Fack and Vanden Berghe [23] and Flessas [75].b Fack et al [78].c Fack and Vanden Berghe [23] and Varma [76].d This result is calculated by using expressions in Fack and Vanden Berghe [23] and Flessas [75].e This result is calculated by using expressions in Fack and Vanden Berghe [23] and Varma [76].f Hodgson [79].

4472H

Ishikawa

Table 10. Eigenvalues of the discretized matrix equation and matrix elements Eν = 〈ν| − (d/dξ)2 + U(ξ)|ν〉/〈ν|ν〉 for a wavefunction with quantumnumber ν of a Morse potential (A) V (x) = V0[exp(−2αx) − 2 exp(−αx)] for depth V0/E0 and (B) V (x) = D[1 − exp(−α(x − xe))]2 by using thecentral-difference integration formula.

Depth Quantum no Eigenvalue Eν Exact

(A) 1.0 0 −2.499 999 999 993 25D−01 −2.500 000 000 000 00D−01 −0.252.25 0 −1.000 000 000 000 78D+00 −1.000 000 000 000 00D+00 −1.06.25 0 −4.000 000 000 000 18D+00 −4.000 000 000 000 05D+00 −4.0

1 −1.000 000 000 000 18D+00 −1.000 000 000 000 10D+00 −1.012.25 0 −9.000 000 000 000 14D+00 −9.000 000 000 000 00D+00 −9.0

1 −3.999 999 999 998 92D+00 −4.000 000 000 000 01D+00 −4.02 −1.000 000 000 000 97D+00 −1.000 000 000 000 01D+00 −1.0

(B) 0 2.408 996 799 999 84D+01 2.408 996 800 000 00D+01 2.408 996 800 000 00D+01a

1 7.080 307 200 000 88D+01 7.080 307 199 999 97D+01 7.080 307 200 000 00D+01a

2 1.155 603 999 999 99D+02 1.155 603 999 999 99D+02 1.155 604 000 000 00D+02a

3 1.583 619 519 999 96D+02 1.583 619 519 999 95D+02 1.583 619 520 000 00D+02a

4 1.992 077 279 999 83D+02 1.992 0772 799 998 6D+02 1.992 077 280 000 00D+02a

5 2.380 977 279 999 97D+02 2.380 977 279 999 68D+02 2.380 977 280 000 00D+02a

6 2.750 319 520 000 07D+02 2.750 319 519 999 34D+02 2.750 319 520 000 00D+02a

7 3.100 104 000 000 01D+02 3.100 103 999 998 81D+02 3.100 104 000 000 00D+02a

8 3.430 330 719 999 89D+02 3.430 330 719 998 04D+02 3.430 330 720 000 00D+02a

9 3.740 999 680 000 06D+02 3.740 999 679 997 00D+02 3.740 999 680 000 00D+02a

10 4.032 110 880 000 00D+02 4.032 110 879 995 71D+02 4.032 110 880 000 00D+02a

a Dagher and Kobeissi [84].


Table 11. Eigenvalues of the discretized matrix equation and matrix elements Eν = 〈ν|−(d/dξ)2 +U(ξ)|ν〉/〈ν|ν〉 for a wavefunction with quantum number ν of a modified Poschl–Teller potentialfor depth V0/E0 by using the central-difference integration formula.

Depth Quantum no. Eigenvalue Eν Exact

1.0 0 −3.819 660 112 500 28D−01 −3.819 660 112 500 98D−01 −3.819 660 112 501 05D−012.0 0 −1.000 000 000 000 22D+00 −1.000 000 000 000 00D+00 −1.06.0 0 −4.000 000 000 000 23D+00 −4.000 000 000 000 02D+00 −4.0

1 −1.000 000 000 002 50D+00 −1.000 000 000 000 04D+00 −1.012.0 0 −9.000 000 000 001 13D+00 −9.000 000 000 000 15D+00 −9.0

1 −4.000 000 000 000 68D+00 −4.000 000 000 000 55D+00 −4.02 −1.000 000 000 001 58D−01 −1.000 000 000 000 66D+00 −1.0

larger than or equal to 10, the whole interval (1.119 6675, 6.432 1675) and h = 1128 . In [25] the

relative error for E10 was of the order of 1.0D-8 for degree 10, the whole interval (0.8, 4.96)

and h = 0.01; we obtain the very close result with relative error 3.4D-8 for the same condition.However, by moving the whole interval to (1.12, 5.28) we obtain the relative error 2.8D-11 forE10 and other eigenvalues are also improved. Thus it is important to choose appropriately thewhole interval. In [51] the discretized matrix method of Numerov with defect correction wasused for calculating eigenvalues of the Morse potential. The relative error of the eigenvaluesranged widely, from 5.0D-14 for ν = 0 to 8.6D-10 for ν = 9.

The fifth example is a symmetric nonlinear potential hole of the form U(x) =−V0/ cosh2(αx), where V0 is a constant [3–5]. The potential is symmetric with respect tox = 0 and has a finite value between −V0 and zero. The number of bound states is alsofinite for this potential. The Schrodinger equation can be reduced to dimensionless form byintroducing units of energy E0 = h2/(2mα2) and of length ξ = αx. The eigenvalue for thebound state with quantum number ν is given by

λν = (Eν/E0) = −{−(1 + 2ν) + [1 + 4(V0/E0)]1/2}2/4 (30)

where n = 0, 1, 2, . . . with ν < {−1 + [1 + 4(V0/E0)]1/2}/2, and the wavefunction is givenin [3–5]. We show typical cases for V0/E0 in table 11, where the degree is 14, the wholeinterval and h are (−26.7, 26.7) and 1

24 for V0/E0 = 1.0, while the other corresponding valuesare (−20.0, 20.0) and 1

32 for V0/E0 = 2.0, 6.0 and 12.0. Since the eigenfunction extendswidely for the finite-depth potentials, the whole interval should also be correspondingly wider.The eigenvalues and matrix elements of the Hamiltonian are of 13- to 15-digit accuracy.

5. Conclusion

We have developed a method for accurate numerical calculation of matrix elements inquantum mechanics in one dimension. Increasing the degree of the classical formulae yieldsfruitful results, i.e. high precision for interpolation, derivative, integration and solution of theeigenvalue problem of ordinary differential equations. We believe the method presented hereis the most concise and accurate available.

Acknowledgments

The author thanks Professor Emeritus Shigeyuki Aono at Kanazawa University and Dr ToshiakiIitaka at RIKEN (The Institute of Physical and Chemical Research) for comments on themanuscript.

4474 H Ishikawa

Appendix. Derivation of the central-difference integration formula

Let a function y = f (x) be given in tabular form at discrete and distinct (n + 1) pointsyk = f (xk), centred at xi , k = i − (n/2), i − (n/2) + 1, . . . , i − 1, i, i + 1, . . . , i + (n/2),where n is an even number and xk is arranged in increasing order. The integral over the threecentral points with interval [xi−1, xi+1] is given in terms of the central difference δnfi as∫ xi+1

xi−1

f (x) dx = 2h[fi + 16δ2fi − 1

180δ4fi

+ 11512δ6fi − 23

226 800δ8fi + · · ·]. (A.1)

This formula can be obtained by using the operational method for the central-differenceintegration formula [28, 85] or by integrating Stirling’s formula [9]. Applying the formulaof the central difference [30]

δnfi =n∑

k=0

(−1)k[n!/k!(n − k)!]f i+(n/2)−k (A.2)

to the right-hand side of equation (A.1), we obtain the integration formula containing termsup to δnfi for n = 6 and 8:∫ xi+1

xi−1

f (x) dx = (h/3780)[5fi−3 − 72fi−2 + 1503fi−1 + 4688fi

+ 1503fi+1 − 72fi+2 + 5fi+3] + O(h9) (A.3)∫ xi+1

xi−1

f (x) dx = (h/113400)[−23fi−4 + 334fi−3 − 2804fi−2 + 46378fi−1

+ 139 030fi + 463 78fi+1 − 2804fi+2 + 334fi+3 − 23fi+4] + O(h11). (A.4)

The formulae for n = 2 and 4 are shown in [9].

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